Models of free quantum field theories on curved backgrounds

TL;DR

This paper explores free quantum field theories on curved spacetimes, detailing their classical and quantum structures through explicit examples of scalar, Dirac, and Proca fields, emphasizing the algebraic approach and properties of globally hyperbolic manifolds.

Contribution

It provides a systematic construction of quantum field models on curved backgrounds using algebraic methods and explicit examples, clarifying the role of globally hyperbolic spacetimes.

Findings

Construction of classical solution spaces for free fields

Development of a $*$-algebra encoding quantum dynamics

Explicit examples of scalar, Dirac, and Proca fields on curved backgrounds

Abstract

Free quantum field theories on curved backgrounds are discussed via three explicit examples: the real scalar field, the Dirac field and the Proca field. The first step consists of outlining the main properties of globally hyperbolic spacetimes, that is the class of manifolds on which the classical dynamics of all physically relevant free fields can be written in terms of a Cauchy problem. The set of all smooth solutions of the latter encompasses the dynamically allowed configurations which are used to identify via a suitable pairing a collection of classical observables. As a last step we use such collection to construct a $*$-algebra which encodes the information on the dynamics and on the canonical commutation or anti-commutation relations depending whether the underlying field is a Fermion or a Boson.